NEW MULTIDIMENSIONAL FUNCTIONAL DIVERSITY INDICES FOR … · Moreover, the search for the effects of biotic interactions and environmental ﬁlters on biodiversity patterns may beneﬁt

Ecology, 89(8), 2008, pp. 2290–2301� 2008 by the Ecological Society of America

NEW MULTIDIMENSIONAL FUNCTIONAL DIVERSITY INDICESFOR A MULTIFACETED FRAMEWORK IN FUNCTIONAL ECOLOGY

SEBASTIEN VILLEGER,1 NORMAN W. H. MASON,2 AND DAVID MOUILLOT1,3

1UMR CNRS-IFREMER-UM2 5119 Ecosystemes Lagunaires, Universite Montpellier 2 CC 093,34 095 Montpellier Cedex 5 France

2Unite de Recherche Hydrobiologie, CEMAGREF, 361 rue Jean-Francois Breton, BP 5095, 34196 Montpellier Cedex 5 France

Abstract. Functional diversity is increasingly identified as an important driver ofecosystem functioning. Various indices have been proposed to measure the functionaldiversity of a community, but there is still no consensus on which are most suitable. Indeed,none of the existing indices meets all the criteria required for general use. The main criteria arethat they must be designed to deal with several traits, take into account abundances, andmeasure all the facets of functional diversity. Here we propose three indices to quantify eachfacet of functional diversity for a community with species distributed in a multidimensionalfunctional space: functional richness (volume of the functional space occupied by thecommunity), functional evenness (regularity of the distribution of abundance in this volume),and functional divergence (divergence in the distribution of abundance in this volume).Functional richness is estimated using the existing convex hull volume index. The newfunctional evenness index is based on the minimum spanning tree which links all the species inthe multidimensional functional space. Then this new index quantifies the regularity withwhich species abundances are distributed along the spanning tree. Functional divergence ismeasured using a novel index which quantifies how species diverge in their distances (weightedby their abundance) from the center of gravity in the functional space. We show that none ofthe indices meets all the criteria required for a functional diversity index, but instead we showthat the set of three complementary indices meets these criteria. Through simulations ofartificial data sets, we demonstrate that functional divergence and functional evenness areindependent of species richness and that the three functional diversity indices are independentof each other. Overall, our study suggests that decomposition of functional diversity into itsthree primary components provides a meaningful framework for its quantification and for theclassification of existing functional diversity indices. This decomposition has the potential toshed light on the role of biodiversity on ecosystem functioning and on the influence of bioticand abiotic filters on the structure of species communities. Finally, we propose a generalframework for applying these three functional diversity indices.

Key words: competitive filtering; environmental filtering; functional divergence; functional evenness;functional niche; functional richness; functional traits; null model.

INTRODUCTION

The functional diversity of a community has emerged

as a facet of biodiversity quantifying the value and range

of organismal traits that influence their performance and

thus ecosystem functioning (Diaz and Cabido 2001).

There is an increasing body of literature demonstrating

that functional diversity, rather than species diversity,

enhances ecosystem functions such as productivity

(Tilman et al. 1997, Hooper and Dukes 2004, Petchey

et al. 2004, Hooper et al. 2005), resilience to perturba-

tions or invasion (Dukes 2001, Bellwood et al. 2004),

and regulation in the flux of matter (Waldbusser et al.

2004). However, most of this work has used functional-

group richness as a surrogate for functional diversity.

Gathering species into groups results in the loss of

information and the imposition of a discrete structure

on functional differences between species, which are

usually continuous (Gitay and Noble 1997, Fonseca and

Ganade 2001). Further, most studies using functional

groups ignore species abundances, and some species may

have a much greater impact on ecosystem functioning

because of their greater abundance (Diaz and Cabido

2001). Finally, a functional-group approach may pro-

duce different conclusions on the importance of

functional diversity for ecosystem functioning depend-

ing on the classification method employed (Wright et al.

2006). Thus, there is an urgent need to provide

continuous measures of functional diversity that directly

use quantitative values for functional traits.

Since 1999, many indices of functional diversity have

been published (reviewed in Petchey and Gaston 2006).

These indices include a priori classifications (number of

functional groups), the sum (FAD;Walker et al. 1999) or

Manuscript received 24 July 2007; revised 21 December 2007;accepted 2 January 2008. Corresponding Editor: N. J. Gotelli.

3 Corresponding author.E-mail: [email protected]

2290

average (quadratic entropy; Botta-Dukat 2005) of

functional distances between species pairs in multivariate

functional trait space, distances between species along

hierarchical classifications (FD; Petchey and Gaston

2002), and the distribution of abundance along func-

tional trait axes (FDvar; Mason et al. 2003). However,

despite the number and variety of these indices, no

consensus has arisen on the sensitive question of how to

measure functional diversity (Petchey and Gaston 2006).

Existing indices have four main limitations:

1) None but FDvar (Mason et al. 2003), the

functional regularity index (FRO) of Mouillot et al.

(2005), and quadratic entropy (Botta-Dukat 2005) take

into account the relative abundance of species. However,

as suggested by Grime (1998), the effect of each species

has to be weighted according to its abundance in order

to reflect its contribution to ecosystem functioning.

2) Some indices are only designed for single-trait

approaches (Mason et al. 2005) and as such may give an

incomplete image of functional diversity when many

traits are used to characterize species functional niches.

3) Some indices are trivially related to species richness,

especially the FAD of Walker et al. (1999), where the

addition of a new species that is completely identical

functionally to another one already in the community

causes an augmentation of functional diversity.

4) While indices based on sum of lengths on

hierarchical classifications do not take account of

abundances, they may deal with many traits simulta-

neously and are not trivially related to species richness

(e.g., the FD index of Petchey and Gaston 2002).

Nevertheless, building a classification from the matrix of

distances between species pairs leads to a loss of

information and modifies the initial interspecific func-

tional distances (as demonstrated by Podani and

Schmera [2006]). In other words, a species classification

cannot match exactly the relative position of species in a

multidimensional functional trait space, and the arbi-

trary choices in the way of constructing classifications

may drastically influence the functional diversity esti-

mation (Podani and Schmera [2007]; but see Petchey and

Gaston [2007]).

Recently, Mason et al. (2005) argued that functional

diversity cannot be summarized by a single number.

Instead they proposed a framework where functional

diversity is composed of three independent compo-

nents—functional richness, functional evenness, and

functional divergence—which need to be quantified

separately. The interest of splitting functional diversity

into three independent components is to provide more

detail in examining the mechanisms linking biodiversity

to ecosystem functioning. For example, Mason et al.

(2008) demonstrated how the primary functional diver-

sity components could be used in combination to test

competing hypotheses for species–energy relationships.

Moreover, the search for the effects of biotic interactions

and environmental filters on biodiversity patterns may

benefit from the proposition of such independent

‘‘facets’’ of functional diversity, since variation in the

volume of functional-trait space filled by species does

not have the same meaning as a shift in the distribution

of abundance within that space. The former may

indicate an increasing pressure of environmental filters

(Cornwell et al. 2006), while the latter may reveal a shift

in the intensity of competitive interactions (Mason et al.

2007, 2008).

However, the indices proposed by Mason et al. (2005)

are estimated based on single traits and are not directly

transposable for multiple-trait approaches. Here we aim

to estimate the three primary components of functional

diversity using one existing and two novel indices that

are specifically designed to incorporate multiple func-

tional traits. These indices directly measure the distri-

bution of species in multivariate functional trait space

and are independent of species richness and each other.

We present these indices as general tools for quantifying

the functional diversity of any community and propose a

practical framework for their application. This frame-

work is based on the use of null models to allow

comparison of communities from different species pools

and with different species richness. It is hoped this

framework will aid in exploring biodiversity–environ-

ment–ecosystem functioning relationships in ecology, as

well as elucidating processes of community assembly

(e.g., Mason et al. 2008).

MULTIDIMENSIONAL FUNCTIONAL DIVERSITY INDICES

As proposed by Keddy (1992), functional ecologists

generally have, for their communities of interest, a

matrix with values for selected functional traits for each

species. From a geometrical point of view, a species’

functional niche may be described by its position in a

functional-trait space (Rosenfeld 2002). Assuming that

we have T functional-trait values for each species of a

given community, the functional-niche space is then the

T dimensional space defined by the T axes, each one

corresponding to a trait.

We suggest standardizing trait values (mean of 0 and

unit variance) so that each trait has the same weight in

functional diversity estimation and the units used to

measure traits have no influence. The community

studied is composed of S species. Any species i has T

traits of standardized values (xi1, xi2, . . . , xiT) which are

conceived as coordinates in the functional trait space.

When plotting all the S species in a multi-trait space,

functional diversity is simply the distribution of species

and their abundances in this functional space (Fig. 1a,

circles represent species, and diameters are species

relative abundances). The indices presented here aim at

describing how much space is filled and how the

abundance of a community is distributed within this

functional space. In the following paragraphs we will

keep this general framework of S species plotted in a T

dimensional space.

Relative abundances of species are noted (w1, w2, . . .

wS), with RSi¼1 wi ¼ 1.

August 2008 2291MULTIDIMENSIONAL FUNCTIONAL DIVERSITY

Following Grime (1998), who underlined the biomass

ratio effect, we suggest that good sets of functional

diversity indices have to take into account biomass, or at

least another estimation of abundance (e.g., number of

individuals, percent cover, or density). Indeed, biomass

is directly linked to the amount of energy and resources

assimilated within a species. Hence, we prefer this

measure of abundance even if the indices may incorpo-

rate any measure of abundance, since they are based on

relative abundances, which are by definition unitless. All

of our indices are also suitable for presence/absence

data, which is actually a particular case where each

species has a relative abundance of 1/S. We will give a

brief description of each primary functional diversity

component and outline the indices we propose for their

measurement in multivariate functional trait space.

Functional richness

Functional richness represents the amount of func-

tional space filled by the community. For a single-trait

approach, the functional richness may be estimated as

the difference between the maximum and minimum

functional values present in the community (Mason et

al. 2005). For multiple-trait studies, functional richness

is more challenging to measure, as the index has to

estimate the volume filled in the T dimensional space by

the community of interest. Recently, Cornwell et al.

(2006) proposed the convex hull volume as a measure of

the functional space occupied by a community. The

convex hull is actually the minimum convex hull which

includes all the species considered; the convex hull

volume is then the volume inside this hull (Fig. 1b).

Thus, if two species a and b are inside the convex hull

volume, whose coordinates (i.e., traits values) are

respectively (xa1, xa2, . . . xaT) and (xb1, xb2, . . . xbT),

then any hypothetical species with coordinates (Kxa1 þ(1� K)xb1, Kxa2þ (1� K)xb2, . . ., KxaTþ (1� K)xbT) for

0 � K � 1 is also in the convex hull volume. This

measure of space occupancy corresponds to a multivar-

iate range. Any species whose trait values are less

extreme for all traits than those of the two existing

species will be included inside the convex hull volume.

FIG. 1. Estimation of the three functional diversity indices in multidimensional functional space. For simplification, only twotraits and nine species are considered. (a) The points are plotted in the space according to the trait values of the correspondingspecies. Circle diameters are proportional to species abundances. In (b), the convex hull is drawn with a solid black line; the pointscorresponding to the vertices are black, and the convex hull volume is shaded in gray. The functional richness (FRic) correspondsto this volume. (c) The minimum spanning tree (MST, dashed line) links the points. Functional evenness (FEve) measures theregularity of points along this tree and the regularity in their abundances. For convenience, the tree is plotted stretched under thepanel. (d) The position of the center of gravity of the vertices (‘‘GV,’’ black cross), the distances between it and the pointsrepresenting the species (gray dashed lines), and the mean distance to the center of gravity (large circle with the black line border).The deviation of the distances from the mean corresponds to the length of the black line linking each point and the large circle withthe black line border. This distribution is also represented under the panel. The more the high abundances are greater than themean, the higher the functional divergence (FDiv).

SEBASTIEN VILLEGER ET AL.2292 Ecology, Vol. 89, No. 8

Basically, the convex hull volume algorithm determines

the most extreme points (hereafter named vertices, the

black circles on Fig. 1b), links them to build the convexhull (lines on Fig. 1b), and finally calculates the volume

inside. Therefore, we propose to use the value of the

convex hull volume filled by a community as amultidimensional measure of the functional richness

(Cornwell et al. 2006, Layman et al. 2007).

We suggest computing the convex hull volume with

the Quickhull algorithm (Barber et al. 1996). Thenumber of species must be higher than the number of

traits (S . T ), and the species must not be distributed in

a line (in which case the hull volume is zero). Theprogram returns the volume and the identity of the

species forming the vertices.

Functional evenness

Functional evenness describes the evenness of abun-

dance distribution in a functional trait space (Mason et

al. 2005). The functional regularity index (FRO) hasbeen proposed for the estimation of functional evenness

when using a single trait (Mouillot et al. 2005). This

index measures both the regularity of spacing betweenspecies along a functional trait gradient and evenness in

the distribution of abundance across species. FRO takes

a value of 1 when the distances between all nearest

neighbor species pairs are identical and when all specieshave the same abundance. Conversely, FRO will

approach 0 when some species are tightly packed along

the functional axis, with a high proportion of abundanceconcentrated within a narrow part of the functional-trait

gradient. It has the advantage of being independent

from species richness, functional richness, and function-al divergence. While an extension to multiple trait

studies has been proposed for FRO (Mouillot et al.

2005), this method is dependent on ordination tech-

niques and consequently risks the loss of information,especially for traits that are weakly correlated with other

traits.

In order to transform species distribution in a T-

dimensional functional space to a distribution on asingle axis, we choose to use the minimum spanning tree

(noted MST hereafter). The MST is the tree that links all

the points contained in a T-dimensional space with theminimum sum of branch lengths (Fig. 1d). We compute

the MST thanks to the ‘‘ape’’ R-package, which returns

the S � 1 branches between the S species. By directanalogy to Mouillot et al. (2005), our new functional

evenness index measures both the regularity of branch

lengths in the MST and evenness in species abundances.

As a first step, for each branch l of the MST (dashed lineon Fig. 1d), the length is divided by the sum of the

abundances of the two species linked by the branch

EWl ¼distði; jÞwi þ wj

where EW is weighted evenness, dist(i, j ) is the

Euclidean distance between species i and j, the species

involved is branch l, and wi is the relative abundance of

species i.

Then, for each of these branches, the value of EWl isdivided by the sum of EW values for the MST to obtain

the partial weighted evenness (PEW), defined as

PEWl ¼EWl

XS�1

l¼1

EWl

:

In the case of perfect regularity of abundance distribu-

tion along the MST, all EWl will be equal and all PEWl

values will be 1/(S� 1). Conversely, when PEWl values

differ among branches, the final index must decrease. To

this aim we compared PEWl values to 1/(S� 1). Finally,our functional evenness index is

FEve ¼

XS�1

l¼1

min PEWl;1

S� 1

� �� 1

S� 1

1� 1

S� 1

:

The term 1/(S� 1) is subtracted from the numerator anddenominator because there is at least one value of PEWl

which is less than or equal to 1/(S� 1) whatever S is (see

Bulla [1994] for more details about this standardiza-tion). Therefore, FEve is not biased by species richness

and is constrained between 0 and 1. We obtain 1 when

all PEWl are equal to 1/(S � 1). FEve is alsoindependent of the convex hull volume, as it is unitless.

We need at least three species to define an MST and

then estimate FEve.Basically, the new index quantifies the regularity with

which the functional space is filled by species, weighted

by their abundance. FEve decreases either whenabundance is less evenly distributed among species or

when functional distances among species are less regular

(Fig. 2).

Functional divergence

For a single-trait approach, functional divergencerepresents how abundance is spread along a functional

trait axis, within the range occupied by the community

(Mason et al. 2005). For instance, divergence is lowwhen the most abundant species have functional traits

that are close to the center of the functional trait range.

Conversely, when the most abundant species haveextreme functional trait values, then divergence is high.

In a multivariate context, functional divergence relatesto how abundance is distributed within the volume of

functional trait space occupied by species (Fig. 1c).

One complication is finding a method to measure

functional divergence that is independent of the volumeof functional trait space occupied and the evenness of

abundance distribution with that volume. Here we

present a novel index to achieve this. Firstly, thecoordinates of the center of gravity GV (g1, g2, . . . gT)

of the V species forming the vertices of the convex hull

are calculated as follows:


gk ¼1

V

XV

i¼1

xik

where xik is the coordinate of species i on trait k [1, T ].

Second, for each of the S species, we calculate the

Euclidean distance to this center of gravity:

dGi ¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiXT

k¼1

ðxik � gkÞ2vuut :

The mean distance of the S species to the center of

gravity (dG) is then calculated:

dG ¼ 1

S

XS

i¼1

dGi:

It is important to note that the coordinates of the

center of gravity are calculated only on vertices

coordinates without taking into account relative abun-

dances; this implies that dGi and thus dG values are only

influenced by the shape and the volume of the scatter

plot of the S species (Fig. 2).

Then, the sum of abundance-weighted deviances (Dd )

and absolute abundance-weighted deviances (Djdj) for

distances from the center of gravity are calculated across

the species:

Dd ¼XS

i¼1

wi 3ðdGi � dGÞ

and

FIG. 2. The center panel (a) is the reference showing that there are nine species and two traits. The figure shows the effect ofchanges in (d, e) the identities of species and (b, c) their relative abundance on functional divergence (FDiv) and functional evenness(FEve). The key to symbols and lines is the same as for Fig. 1. For simplicity, the coordinates of the extreme points are the same forall the figures, as are functional richness values (FRic) and the shape of the convex hull.


Djdj ¼XS

i¼1

wi 3 jdGi � dGj:

Functional divergence may then be calculated as

FDiv ¼ Dd þ dG

Djdj þ dG:

Values of dGi are Euclidean distances and thus are

positive or null, hence Dd is bounded between dG and

Djdj. Therefore, addition of dG to the numerator and

denominator ensures that the index ranges between 0

and 1. The index approaches 0 when highly abundant

species are very close to the center of gravity relative to

rare species (Dd is negative and tends to �dG), and it

approaches unity when highly abundant species are very

distant from the center of gravity relative to rare species

(Dd is positive and tends to Djdj; see Fig. 2 for

illustration).

For presence/absence data, functional divergence is

the highest if all the species are on the convex hull and at

equal distance to its center of gravity (i.e., if the center of

gravity of the convex hull is also a center of symmetry of

the functional space). This condition is actually true

whatever the relative abundances of species.

Given that the distances considered in the formula are

those from the center of gravity of the vertices,

functional divergence is a priori independent from the

shape and the volume of the convex hull and so from the

functional richness index. A script (R statistical lan-

guage [R development Core Team 2007]) to compute the

three indices is available online.4

ASSESSING THE VALIDITY OF THE INDICES

Some authors have proposed criteria that functional

diversity indices have to match (Mason et al. 2003,

Ricotta et al. 2005, Petchey and Gaston 2006). From

these published criteria we selected 10 that appear

relevant for a multidimensional approach (and which

are not contradictory). We do not expect that each of

our three indices matches each criterion but rather that

the ensemble of indices does (Table 1).

First, our three indices are positive, and the higher

they are, the higher the component of functional

diversity they quantify is. Functional divergence (FDiv)

and functional evenness (FEve) are strictly constrained

between 0 and 1. Functional richness has no upper limit

because it quantifies an absolute volume filled, which

depends partly on the number of traits and on their unit.

However, functional richness values may be constrained

between 0 and 1 via standardization by the global hull

volume (e.g., the volume occupied by all species

considered in a particular study, as proposed for the

FRi index of Mason et al. 2005). The three indices are

independent of the unit used to measure species

abundances. Indeed, functional richness is, by construc-

tion, independent of species abundances while function-

al divergence and functional evenness take into account

species relative abundances, which are unitless. Func-

tional divergence and functional evenness both reflect

the distribution of species abundances in functional

space, and thus the contribution of each species to

functional divergence and functional evenness is pro-

portional to its abundance.

Functional richness is the only index that reflects the

range of the trait values and thus, as expected, is affected

by the unit of the traits (Fig. 3). However, as noted

previously, this may be accounted through standardiza-

tion by the maximum possible hull volume. The three

indices are not affected by a translation or a rotation of

the functional space of reference (Fig. 3).

To test whether our indices match the criteria of

independence with species richness and whether our

indices are independent from each other, we generated

artificial communities. The number of traits was fixed to

three. Coordinates of the species for each axis were

generated using a uniform distribution (i.e., all values

had equal chance of being selected) within a range of 10.

Seven species richness values were considered (10, 15, 20,

25, 30, 35, and 40). Species abundances were generated

TABLE 1. Summary of the criteria of Mason et al. (2003) and Ricotta (2005) and properties of our three indices (FRic forfunctional richness, FDiv for functional divergence, and FEve for functional evenness) for these criteria.

Criteria FRic FEve FDiv

Positive values� yes yes yesBe constrained to a 0–1 range (for convenience) and use that range well� yes yesBe unaffected by the units in which the abundance is measured� yes yes yesReflect the contribution of each species in proportion to its abundance� yes yesBe unaffected by the units in which the character is measured� yes yesReflect the range of character values present� yesBe unaffected by the number of species� yes yesBe unaffected when a species is split in two species with the same traits valuesand the same total abundance�

yes yes

Set monotonicity (a subset of a community is less diverse that this community)� yesConcavity (a set of communities is in mean less diverse than the aggregated pool)� yes

Note: Empty cells did not meet the criteria of that particular index.� These criteria are from Ricotta (2005).� These criteria are from Mason et al. (2003).

4 hhttp://www.ecolag.univ-montp2.fr/software/i


using a uniform distribution within a range of 100 and

then standardized to relative abundances. One hundred

replicates (coordinates and abundances) were generated

using R software for each of these seven cases. The three

indices were computed for these 700 data sets. Pearson’s

coefficients of correlation between each index and

species richness and between the three indices were then

tested.

Our simulations using artificial data sets showed

clearly that functional richness and species richness are

strongly and positively related (r¼0.872, P , 0.001; Fig.

4a). As expected from the sampling effect, it is more

likely to obtain a larger hull volume with more species in

the community. However, for a given T-dimensional

functional space, the maximal convex hull is obtained

with 2T species whose coordinates are a combination of

the extreme values on each axis. In other words, in such

a given Euclidian space, the maximal volume given the

ranges on the axes is a hypervolume with all its angles

square. For example, with a three-dimensional space,

the maximum convex hull volume is obtained with at

least eight points constituting a cube. It is impossible to

fill the whole available space with fewer than eight

species. This property means that for communities with

species richness less than 2T, observed functional

richness values are not comparable. Practically, to avoid

this bias, the number of species must increase exponen-

tially with the number of traits in comparative studies.

Cornwell et al. (2006) do not explicitly point out this

bias in their study, but they overcome it using a

randomization procedure allowing the observed func-

tional richness value to be compared with that expected

at random for the species richness level of the

community. This limit partly explains the positive

correlation between the convex hull volume index and

species richness, the shape of the relation depending on

the number of traits, and on their correlations.

FIG. 3. Properties of the three functional diversity indices (FRic for functional richness, FDiv for functional divergence, andFEve for functional evenness). Two traits (axes) and nine species (points) with equal abundances are considered for graphicalcommodity. The key to the symbols and lines is the same as for Fig. 1. Panel (a) is the community of reference. Two communitieswith (c) a rotation or with (d) a translation between them have similar functional richness, functional evenness, and functionaldivergence. However, (b) two communities having the same shape but different sizes have similar functional evenness andfunctional divergence while having different functional richness values.


The values obtained with artificial communities

showed that functional divergence and functional

evenness are independent from species richness (Fig.

4b, c). These two indices are also independent from

functional richness (Fig. 4d, e) and each other (Fig. 4f ).

Two of the three indices satisfied the modified

twinning criterion proposed by Mason et al. (2003),

that diversity is not affected when a species is replaced

by two species with the same trait values and the same

total abundance. If the species is a vertex, the convex

hull volume algorithm will consider only one of the two

twins as a vertex and thus the convex hull, and so the

richness will not be modified. Similarly, the position of

the center of gravity of the vertices will not be affected.

Moreover, the Dd and Djdj of the FDiv computation will

be unchanged by the fact that the abundance is split

between two entities having the same position. On the

contrary, the evenness index does not satisfy this

criterion. If a species is split into two species identical

for all traits and which share the initial abundance, the

regularity of trait values will be changed, as there are

now two species at the same place in the functional

space. It is not a bias of our index but is inherent to our

definition of functional evenness. This would only be

problematic when it is difficult to identify species with

certainty or where taxonomic opinion differs as to

whether a subspecific taxon should be treated as a

separate species. Thus, we believe that the latter point,

indices independent from species richness, is far more

important in the ecological context.

Functional richness is the only index in accordance

with the monotonicity criterion proposed by Ricotta et

al. (2005). Indeed, the functional-richness value of a

subset of species cannot be superior to the functional-

richness value of the whole set. Functional divergence

and functional evenness do not match this criterion, as

the addition of species can decrease functional diver-

gence (a new abundant species close to the center of

gravity of the functional space) and functional evenness

(a new species close to an abundant species), because

these indices consider relative abundances.

Similarly, functional richness is the only index to

respect the concavity criterion, for the same reason as

for the monotonicity criterion. By contrast, indices of

divergence and evenness are not additive, which means

that the divergence (or the evenness) of two communities

is not linked to the mean of the two index values but

depends on the characteristics of the new constructed

community (species traits and relative abundances).

FIG. 4. Properties of the three functional diversity indices for artificial communities. Three traits were considered, and both thecoordinates and the abundances of the species were generated under a uniform law (with respective range of 10 and 100). Sevenspecies richness levels (S) were considered. Each species richness level was replicated 100 times. For each community, functionalrichness (FRic), functional divergence (FDiv), and functional evenness (FEve) were estimated. The first three panels (a, b, c) showthe relations between each index and species richness. The three last panels (d, e, f ) present the correlations between the threeindices. Pearson’s coefficients of correlation and levels of significance are given above the panels. FRic is the only index correlatedto species richness. The three indices are independent of each other.


In summary, our set of three indices meets all the

criteria required for functional diversity measures. The

three indices are actually complementary. Moreover,

functional divergence and evenness are independent

from species richness, which allows comparison of

communities with different species richness without

bias. Similarly, their independence from functional

richness allows for testing of differences in functional

divergence or evenness with different functional richness

values.

FROM THEORY TO PRACTICE

Generally, data sets contain two to four of the

following matrices in Fig. 5: (1) a functional trait matrix

(with values for each of the S species, for each of the T-

functional traits), (2) an ‘‘abundance pattern’’ matrix

FIG. 5. General framework to study the effect of environmental conditions on functional diversity or the effect of functionaldiversity on ecosystem properties.


(with the abundances of each species in the C

communities), (3) an ‘‘environmental’’ matrix (with

values for each of the E environmental variables in each

community), and (4) an ‘‘ecosystem properties’’ matrix

(with values for each of the P ecosystem properties such

as productivity, flux of nutrients, or resistance to

perturbation in each of the communities). The main

limiting factor is that functional traits have to be the

same for the communities to be compared. This

emphasizes the need for consensual lists of traits for

each type of organism (plant, terrestrial animal, fishes,

microorganisms).

Moreover, as underlined by Petchey and Gaston

(2006), the a priori selection of traits is often critical. The

main problems concern the number of traits and their

identity. Indeed, the number of traits is linked to the

amount of work needed to measure them on each

species, but also to the functions that are quantified. The

choice of the traits has to be led by the need to describe

each function as well as possible while avoiding

redundancy (i.e., trivial correlations) between them. In

particular, when using functional traits derived from

other traits (e.g., ratios), original traits should not be

used in calculating functional diversity since the original

and derived traits may be trivially related. If traits are

carefully selected, then any correlation between traits in

the species–trait matrix may be considered a relevant

aspect of species distribution in functional trait space.

For example, the competitive, stress tolerant, and

ruderal strategy (CSR) theory of Grime (1974, 2001)

shows that correlations between traits that have no a

priori link reveal major trends in plant functional

strategy. More generally, correlations between traits

may highlight patterns of species aggregation in a

functional space where species separate into functional

groups, whereas this may not be evident when functional

traits are not correlated. Using ordination axes in

calculating functional diversity will obscure these

correlations. Ordinations also risk the loss of informa-

tion, since the ordination axes can only capture a

proportion of the variation in functional trait values

across species. In summary, if functional traits are

selected so that trivial correlations are avoided, any

correlation between traits will represent a relevant aspect

of species distribution in functional-trait space, and

there will be no need to apply ordination techniques to

obtain orthogonal axes. However, we may encounter

constraints in the data which imply the use of ordination

techniques, such as the use of too many traits compared

to species number or the use of categorical data.

All of our indices are designed to quantify functional

diversity using continuous traits. However, as exposed

by Podani and Schmera (2006), ecological variables and,

in particular, functional traits are sometimes qualitative

either categorical (type of photosynthesis, ability to

sprout after fire) or circular (time of reproduction). To

overcome this problem, we propose to estimate a

distance matrix using distances such the Gower distance

which allows mixing qualitative and quantitative traits

(Podani and Schmera 2006). In this case, a Principal

Coordinates Analysis (PCoA) may be used to represent

species distribution in a multidimensional functional

space. PCoA works on distance matrix and its outputs

are similar to those obtained from PCA (Legendre and

Legendre 1998), i.e., the coordinates of species in a

Euclidean functional space with reduced uncorrelated

dimensions. Another particular case is the use of

presence/absence data. Then, functional divergence

and functional evenness have a different meaning in

the sense that they would quantify the relative position

of species within the functional space instead of the

distribution of abundance.

However, we believe that in order to compare

functional diversity values among communities with

different species richness and different regional species

pools, the best way is to consider observed values

relative to those expected at random. Expected values

may be obtained using a matrix swap randomization

(Manly 1995) that maintains species richness of com-

munities and the frequency of occurrence of species in

randomized matrices. In fact, such a correction when

comparing functional diversity of different local com-

munities is necessary for all indices, since species

functional traits in the pool will constrain the range of

functional-diversity values possible. Methodologies for

comparing observed functional diversity values to those

expected by chance were provided in Mason et al. (2007,

2008).

The primary components of functional diversity

identified by Mason et al. (2005) have aided us in

finding a set of orthogonal multivariate functional

diversity indices to give a comprehensive framework

for the quantification of functional diversity in multidi-

mensional functional trait space. It is possible that this

framework does not capture all aspects of functional

diversity, but it remains the sole available method for

classifying functional diversity indices by the aspect of

species distribution in functional trait space that they

measure (cf. the classification system employed by

Petchey and Gaston 2006). In helping to decide on a

set of orthogonal indices, the primary functional

diversity components aid the application of functional

diversity indices in elucidation patterns and processes in

ecological communities.

Functional diversity may act either as (1) an indicator

of the processes governing community assembly (e.g.,

environmental and competitive filtering; Cornwell et al.

2006) and the impact of perturbations (e.g., climate

change, fire, grazing or overfishing) and environmental

gradients on community structure (e.g., Mouillot et al.

2007), or (2) an indicator of ecosystem functions such as

productivity, resilience, and nutrient cycling (e.g.,

Petchey et al. 2004). Concurrent examination of

functional diversity indices representing separate prima-

ry components increases the detail with which we may

examine a variety of hypotheses relating to these dual


roles of functional diversity. For example, Mason et al.

(2008) found that functional evenness, compared to that

expected at random, increased linearly with mean

annual temperature and species richness in French

lacustrine fish communities, while functional richness

and functional divergence showed asymptotic relation-

ships in both cases. These results, considered together,

suggest that increased niche specialization (as opposed

to an increase in the volume of niche space occupied)

with increasing temperature allowed more species to

coexist in high-energy communities. Similarly, an

orthogonal set of indices might allow comparison of

evidence for increased niche specialization or occupied

niche volume as mechanisms for increased productivity

or resilience. Using this approach, functional diversity

indices may in effect be used to test not only whether

niche complementarity enhances ecosystem function,

but which type of complementarity enhances ecosystem

function the most.

Until now, such an approach has been constrained to

the use of univariate functional diversity indices. The

three indices we propose here allow the implementation

of the primary functional diversity components in

multiple dimensions. They provide independent infor-

mation about the position and relative abundances of

species in a multidimensional functional space. There-

fore, we believe that these new indices may help in the

exploration of biodiversity–environment–ecosystem

functioning relationships in ecology.

ACKNOWLEDGMENTS

This study was partially funded by a LITEAU III (PAMPA)and two ANR (GAIUS and AMPHORE) financial supports tostudy ecological indicators. We thank one anonymous reviewerand J. Podani for their comments and constructive criticisms onour manuscript.

LITERATURE CITED

Barber, C. B., D. P. Dobkin, and H. T. Huhdanpaa. 1996. TheQuickhull algorithm for convex hulls. ACM Transactions onMathematical Software 22:469–483.

Bellwood, D. R., T. P. Hughes, C. Folke, and M. Nystrom.2004. Confronting the coral reef crisis. Nature 429:827–833.

Botta-Dukat, Z. 2005. Rao’s quadratic entropy as a measure offunctional diversity based on multiple traits. Journal ofVegetation Science 16:533–540.

Bulla, L. 1994. An index of evenness and its associated diversitymeasure. Oikos 70:167–171.

Cornwell, W. K., D. W. Schwilk, and D. D. Ackerly. 2006. Atrait-based test for habitat filtering: convex hull volume.Ecology 87:1465–1471.

Diaz, S., and M. Cabido. 2001. Vive la difference: plantfunctional diversity matters to ecosystem processes. Trends inEcology and Evolution 16:646–655.

Dukes, J. S. 2001. Biodiversity and invasibility in grasslandmicrocosms. Oecologia 126:563–568.

Fonseca, C. R., and G. Ganade. 2001. Species functionalredundancy, random extinctions and the stability of ecosys-tems. Journal of Ecology 89:118–125.

Gitay, H., and I. R. Noble. 1997. What are functional types andhow should we seek them? Pages 3–19 in T. M. Smith, H. H.Shugart, and F. I. Woodward, editors. Plant functional types:their relevance to ecosystem properties and global change.Cambridge University Press, Cambridge, UK.

Grime, J. P. 1974. Vegetation classification by reference tostrategies. Nature 250:26–31.

Grime, J. P. 1998. Benefits of plant diversity to ecosystems:immediate, filter and founder effects. Journal of Ecology 86:902–910.

Grime, J. P. 2001. Plant strategies, vegetation processes, andecosystem properties. John Wiley and Sons, New York, NewYork, USA.

Hooper, D. U., and J. S. Dukes. 2004. Overyielding amongplant functional groups in a long-term experiment. EcologyLetters 7:95–105.

Hooper, D. U., et al. 2005. Effects of biodiversity on ecosystemfunctioning: a consensus of current knowledge. EcologicalMonographs 75:3–35.

Keddy, P. A. 1992. Assembly and response rules—2 goals forpredictive community ecology. Journal of Vegetation Science3:157–164.

Layman, C. A., D. A. Arrington, C. G. Montana, and D. M.Post. 2007. Can stable isotope ratios provide for community-wide measures of trophic structure? Ecology 88:42–48.

Legendre, P., and L. Legendre. 1998. Numerical ecology.Elsevier, Amsterdam, The Netherlands.

Manly, B. F. J. 1995. A note on the analysis of species co-occurrences. Ecology 76:1109–1115.

Mason, N. W. H., P. Irz, C. Lanoiselee, D. Mouillot, and C.Argillier. 2008. Evidence that niche specialisation explainsspecies–energy relationships in lake fish communities. Jour-nal of Animal Ecology 77:285–296.

Mason, N. W. H., C. Lanoiselee, D. Mouillot, P. Irz, and C.Argillier. 2007. Functional characters combined with nullmodels reveal inconsistency in mechanisms of speciesturnover in lacustrine fish communities. Oecologia 153:441–452.

Mason, N. W. H., K. MacGillivray, J. B. Steel, and J. B.Wilson. 2003. An index of functional diversity. Journal ofVegetation Science 14:571–578.

Mason, N. W. H., D. Mouillot, W. G. Lee, and J. B. Wilson.2005. Functional richness, functional evenness and functionaldivergence: the primary components of functional diversity.Oikos 111:112–118.

Mouillot, D., O. Dumay, and J. A. Tomasini. 2007. Limitingsimilarity, niche filtering and functional diversity in brackishlagoon fish communities. Estuarine, Coastal and ShelfScience 71:443–456.

Mouillot, D., N. W. H. Mason, O. Dumay, and J. B. Wilson.2005. Functional regularity: a neglected aspect of functionaldiversity. Oecologia 142:353–359.

Petchey, O. L., and K. J. Gaston. 2002. Functional diversity(FD), species richness, and community composition. EcologyLetters 5:402–411.

Petchey, O. L., and K. J. Gaston. 2006. Functional diversity:back to basics and looking forward. Ecology Letters 9:741–758.

Petchey, O. L., and K. J. Gaston. 2007. Dendrograms andmeasuring functional diversity. Oikos 116:1422–1426.

Petchey, O. L., A. Hector, and K. J. Gaston. 2004. How dodifferent measures of functional diversity perform? Ecology85:847–857.

Podani, J., and D. Schmera. 2006. On dendrogram-basedmeasures of functional diversity. Oikos 115:179–185.

Podani, J., and D. Schmera. 2007. How should a dendrogram-based measure of functional diversity function? A rejoinderto Petchey and Gaston. Oikos 116:1427–1430.

R Development Core Team. 2007. R: a language andenvironment for statistical computing. R Foundation forStatistical Computing, Vienna, Austria. hhttp://www.R-project.orgi

Ricotta, C. 2005. A note on functional diversity measures. Basicand Applied Ecology 6:479–486.

Rosenfeld, J. S. 2002. Functional redundancy in ecology andconservation. Oikos 98:156–162.


Tilman, D., J. Knops, D. Weldin, P. Reich, M. Ritchie, and E.Siemann. 1997. The influence of functional diversity andcomposition on ecosystem processes. Science 277:1300–1302.

Waldbusser, G. G., R. L. Marinelli, R. B. Whitlatch, and P. T.Visscher. 2004. The effects of infaunal biodiversity onbiogeochemistry of coastal marine sediments. Limnologyand Oceanography 49:1482–1492.

Walker, B., A. Kinzig, and J. Langridge. 1999. Plant attributediversity, resilience, and ecosystem function: the nature and sig-nificance of dominant and minor species. Ecosystems 2:95–113.

Wright, J. P., S. Naeem, A. Hector, C. Lehman, P. B. Reich, B.Schmid, and D. Tilman. 2006. Conventional functionalclassification schemes underestimate the relationship withecosystem functioning. Ecology Letters 9:111–120.


Documents

NEW MULTIDIMENSIONAL FUNCTIONAL DIVERSITY INDICES FOR … · Moreover, the search for the effects of biotic interactions and environmental ﬁlters on biodiversity patterns may beneﬁt